Properties

Label 507.4.a.m.1.4
Level $507$
Weight $4$
Character 507.1
Self dual yes
Analytic conductor $29.914$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 507.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(29.9139683729\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \( x^{4} - 2x^{3} - 25x^{2} + 24x + 78 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(5.33039\) of defining polynomial
Character \(\chi\) \(=\) 507.1

$q$-expansion

\(f(q)\) \(=\) \(q+5.33039 q^{2} -3.00000 q^{3} +20.4131 q^{4} -16.4131 q^{5} -15.9912 q^{6} +9.67968 q^{7} +66.1667 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+5.33039 q^{2} -3.00000 q^{3} +20.4131 q^{4} -16.4131 q^{5} -15.9912 q^{6} +9.67968 q^{7} +66.1667 q^{8} +9.00000 q^{9} -87.4882 q^{10} +27.5882 q^{11} -61.2393 q^{12} +51.5965 q^{14} +49.2393 q^{15} +189.390 q^{16} +107.928 q^{17} +47.9735 q^{18} -2.24723 q^{19} -335.042 q^{20} -29.0391 q^{21} +147.056 q^{22} +41.8090 q^{23} -198.500 q^{24} +144.390 q^{25} -27.0000 q^{27} +197.592 q^{28} +61.6213 q^{29} +262.465 q^{30} +191.932 q^{31} +480.187 q^{32} -82.7645 q^{33} +575.300 q^{34} -158.874 q^{35} +183.718 q^{36} +98.4236 q^{37} -11.9786 q^{38} -1086.00 q^{40} -30.7452 q^{41} -154.790 q^{42} +238.325 q^{43} +563.160 q^{44} -147.718 q^{45} +222.858 q^{46} -511.482 q^{47} -568.169 q^{48} -249.304 q^{49} +769.653 q^{50} -323.785 q^{51} +492.825 q^{53} -143.921 q^{54} -452.807 q^{55} +640.472 q^{56} +6.74170 q^{57} +328.466 q^{58} +484.179 q^{59} +1005.13 q^{60} -444.021 q^{61} +1023.07 q^{62} +87.1172 q^{63} +1044.47 q^{64} -441.167 q^{66} +190.114 q^{67} +2203.15 q^{68} -125.427 q^{69} -846.858 q^{70} +484.785 q^{71} +595.500 q^{72} -957.780 q^{73} +524.636 q^{74} -433.169 q^{75} -45.8729 q^{76} +267.045 q^{77} -375.216 q^{79} -3108.47 q^{80} +81.0000 q^{81} -163.884 q^{82} -715.765 q^{83} -592.777 q^{84} -1771.43 q^{85} +1270.37 q^{86} -184.864 q^{87} +1825.42 q^{88} -1038.15 q^{89} -787.394 q^{90} +853.451 q^{92} -575.796 q^{93} -2726.40 q^{94} +36.8840 q^{95} -1440.56 q^{96} +65.5636 q^{97} -1328.89 q^{98} +248.293 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 12 q^{3} + 22 q^{4} - 6 q^{5} - 6 q^{6} - 14 q^{7} + 54 q^{8} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 12 q^{3} + 22 q^{4} - 6 q^{5} - 6 q^{6} - 14 q^{7} + 54 q^{8} + 36 q^{9} - 62 q^{10} + 40 q^{11} - 66 q^{12} + 40 q^{14} + 18 q^{15} + 122 q^{16} + 98 q^{17} + 18 q^{18} + 124 q^{19} - 466 q^{20} + 42 q^{21} + 220 q^{22} + 104 q^{23} - 162 q^{24} - 58 q^{25} - 108 q^{27} - 144 q^{28} + 194 q^{29} + 186 q^{30} + 26 q^{31} + 654 q^{32} - 120 q^{33} + 1062 q^{34} + 88 q^{35} + 198 q^{36} + 102 q^{37} + 332 q^{38} - 998 q^{40} - 1054 q^{41} - 120 q^{42} + 450 q^{43} - 44 q^{44} - 54 q^{45} - 172 q^{46} - 96 q^{47} - 366 q^{48} + 1070 q^{49} + 996 q^{50} - 294 q^{51} + 262 q^{53} - 54 q^{54} + 204 q^{55} + 2164 q^{56} - 372 q^{57} + 722 q^{58} + 308 q^{59} + 1398 q^{60} - 928 q^{61} + 2780 q^{62} - 126 q^{63} + 1026 q^{64} - 660 q^{66} - 1134 q^{67} + 1786 q^{68} - 312 q^{69} - 2324 q^{70} + 1064 q^{71} + 486 q^{72} + 952 q^{73} + 1158 q^{74} + 174 q^{75} - 1708 q^{76} + 2508 q^{77} - 746 q^{79} - 2922 q^{80} + 324 q^{81} + 1734 q^{82} - 404 q^{83} + 432 q^{84} - 1394 q^{85} + 3168 q^{86} - 582 q^{87} + 3060 q^{88} + 1620 q^{89} - 558 q^{90} + 332 q^{92} - 78 q^{93} - 772 q^{94} + 2204 q^{95} - 1962 q^{96} + 2166 q^{97} - 1906 q^{98} + 360 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.33039 1.88458 0.942289 0.334800i \(-0.108669\pi\)
0.942289 + 0.334800i \(0.108669\pi\)
\(3\) −3.00000 −0.577350
\(4\) 20.4131 2.55164
\(5\) −16.4131 −1.46803 −0.734016 0.679132i \(-0.762356\pi\)
−0.734016 + 0.679132i \(0.762356\pi\)
\(6\) −15.9912 −1.08806
\(7\) 9.67968 0.522654 0.261327 0.965250i \(-0.415840\pi\)
0.261327 + 0.965250i \(0.415840\pi\)
\(8\) 66.1667 2.92418
\(9\) 9.00000 0.333333
\(10\) −87.4882 −2.76662
\(11\) 27.5882 0.756195 0.378098 0.925766i \(-0.376578\pi\)
0.378098 + 0.925766i \(0.376578\pi\)
\(12\) −61.2393 −1.47319
\(13\) 0 0
\(14\) 51.5965 0.984982
\(15\) 49.2393 0.847568
\(16\) 189.390 2.95921
\(17\) 107.928 1.53979 0.769895 0.638171i \(-0.220309\pi\)
0.769895 + 0.638171i \(0.220309\pi\)
\(18\) 47.9735 0.628193
\(19\) −2.24723 −0.0271342 −0.0135671 0.999908i \(-0.504319\pi\)
−0.0135671 + 0.999908i \(0.504319\pi\)
\(20\) −335.042 −3.74588
\(21\) −29.0391 −0.301754
\(22\) 147.056 1.42511
\(23\) 41.8090 0.379034 0.189517 0.981877i \(-0.439308\pi\)
0.189517 + 0.981877i \(0.439308\pi\)
\(24\) −198.500 −1.68828
\(25\) 144.390 1.15512
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 197.592 1.33362
\(29\) 61.6213 0.394579 0.197289 0.980345i \(-0.436786\pi\)
0.197289 + 0.980345i \(0.436786\pi\)
\(30\) 262.465 1.59731
\(31\) 191.932 1.11200 0.556000 0.831182i \(-0.312335\pi\)
0.556000 + 0.831182i \(0.312335\pi\)
\(32\) 480.187 2.65269
\(33\) −82.7645 −0.436589
\(34\) 575.300 2.90185
\(35\) −158.874 −0.767272
\(36\) 183.718 0.850545
\(37\) 98.4236 0.437317 0.218659 0.975801i \(-0.429832\pi\)
0.218659 + 0.975801i \(0.429832\pi\)
\(38\) −11.9786 −0.0511366
\(39\) 0 0
\(40\) −1086.00 −4.29279
\(41\) −30.7452 −0.117112 −0.0585561 0.998284i \(-0.518650\pi\)
−0.0585561 + 0.998284i \(0.518650\pi\)
\(42\) −154.790 −0.568680
\(43\) 238.325 0.845216 0.422608 0.906313i \(-0.361115\pi\)
0.422608 + 0.906313i \(0.361115\pi\)
\(44\) 563.160 1.92953
\(45\) −147.718 −0.489344
\(46\) 222.858 0.714319
\(47\) −511.482 −1.58739 −0.793695 0.608316i \(-0.791845\pi\)
−0.793695 + 0.608316i \(0.791845\pi\)
\(48\) −568.169 −1.70850
\(49\) −249.304 −0.726833
\(50\) 769.653 2.17691
\(51\) −323.785 −0.888998
\(52\) 0 0
\(53\) 492.825 1.27726 0.638630 0.769514i \(-0.279502\pi\)
0.638630 + 0.769514i \(0.279502\pi\)
\(54\) −143.921 −0.362687
\(55\) −452.807 −1.11012
\(56\) 640.472 1.52833
\(57\) 6.74170 0.0156660
\(58\) 328.466 0.743615
\(59\) 484.179 1.06838 0.534192 0.845363i \(-0.320616\pi\)
0.534192 + 0.845363i \(0.320616\pi\)
\(60\) 1005.13 2.16269
\(61\) −444.021 −0.931985 −0.465993 0.884789i \(-0.654303\pi\)
−0.465993 + 0.884789i \(0.654303\pi\)
\(62\) 1023.07 2.09565
\(63\) 87.1172 0.174218
\(64\) 1044.47 2.03998
\(65\) 0 0
\(66\) −441.167 −0.822787
\(67\) 190.114 0.346658 0.173329 0.984864i \(-0.444548\pi\)
0.173329 + 0.984864i \(0.444548\pi\)
\(68\) 2203.15 3.92898
\(69\) −125.427 −0.218835
\(70\) −846.858 −1.44598
\(71\) 484.785 0.810329 0.405164 0.914244i \(-0.367214\pi\)
0.405164 + 0.914244i \(0.367214\pi\)
\(72\) 595.500 0.974727
\(73\) −957.780 −1.53561 −0.767806 0.640683i \(-0.778651\pi\)
−0.767806 + 0.640683i \(0.778651\pi\)
\(74\) 524.636 0.824159
\(75\) −433.169 −0.666907
\(76\) −45.8729 −0.0692367
\(77\) 267.045 0.395228
\(78\) 0 0
\(79\) −375.216 −0.534368 −0.267184 0.963646i \(-0.586093\pi\)
−0.267184 + 0.963646i \(0.586093\pi\)
\(80\) −3108.47 −4.34422
\(81\) 81.0000 0.111111
\(82\) −163.884 −0.220707
\(83\) −715.765 −0.946571 −0.473286 0.880909i \(-0.656932\pi\)
−0.473286 + 0.880909i \(0.656932\pi\)
\(84\) −592.777 −0.769967
\(85\) −1771.43 −2.26046
\(86\) 1270.37 1.59288
\(87\) −184.864 −0.227810
\(88\) 1825.42 2.21125
\(89\) −1038.15 −1.23645 −0.618224 0.786002i \(-0.712148\pi\)
−0.618224 + 0.786002i \(0.712148\pi\)
\(90\) −787.394 −0.922207
\(91\) 0 0
\(92\) 853.451 0.967157
\(93\) −575.796 −0.642013
\(94\) −2726.40 −2.99156
\(95\) 36.8840 0.0398339
\(96\) −1440.56 −1.53153
\(97\) 65.5636 0.0686286 0.0343143 0.999411i \(-0.489075\pi\)
0.0343143 + 0.999411i \(0.489075\pi\)
\(98\) −1328.89 −1.36977
\(99\) 248.293 0.252065
\(100\) 2947.44 2.94744
\(101\) 531.798 0.523920 0.261960 0.965079i \(-0.415631\pi\)
0.261960 + 0.965079i \(0.415631\pi\)
\(102\) −1725.90 −1.67539
\(103\) −735.984 −0.704064 −0.352032 0.935988i \(-0.614509\pi\)
−0.352032 + 0.935988i \(0.614509\pi\)
\(104\) 0 0
\(105\) 476.621 0.442985
\(106\) 2626.95 2.40710
\(107\) −783.265 −0.707673 −0.353837 0.935307i \(-0.615123\pi\)
−0.353837 + 0.935307i \(0.615123\pi\)
\(108\) −551.153 −0.491063
\(109\) −532.339 −0.467788 −0.233894 0.972262i \(-0.575147\pi\)
−0.233894 + 0.972262i \(0.575147\pi\)
\(110\) −2413.64 −2.09210
\(111\) −295.271 −0.252485
\(112\) 1833.23 1.54664
\(113\) −180.589 −0.150340 −0.0751699 0.997171i \(-0.523950\pi\)
−0.0751699 + 0.997171i \(0.523950\pi\)
\(114\) 35.9359 0.0295237
\(115\) −686.215 −0.556434
\(116\) 1257.88 1.00682
\(117\) 0 0
\(118\) 2580.86 2.01346
\(119\) 1044.71 0.804777
\(120\) 3258.00 2.47844
\(121\) −569.893 −0.428169
\(122\) −2366.81 −1.75640
\(123\) 92.2357 0.0676147
\(124\) 3917.92 2.83742
\(125\) −318.242 −0.227716
\(126\) 464.369 0.328327
\(127\) 1431.63 1.00029 0.500146 0.865941i \(-0.333280\pi\)
0.500146 + 0.865941i \(0.333280\pi\)
\(128\) 1725.94 1.19182
\(129\) −714.976 −0.487986
\(130\) 0 0
\(131\) 2067.32 1.37880 0.689400 0.724381i \(-0.257874\pi\)
0.689400 + 0.724381i \(0.257874\pi\)
\(132\) −1689.48 −1.11402
\(133\) −21.7525 −0.0141818
\(134\) 1013.38 0.653304
\(135\) 443.153 0.282523
\(136\) 7141.25 4.50262
\(137\) −387.512 −0.241660 −0.120830 0.992673i \(-0.538556\pi\)
−0.120830 + 0.992673i \(0.538556\pi\)
\(138\) −668.575 −0.412412
\(139\) 752.568 0.459223 0.229611 0.973282i \(-0.426254\pi\)
0.229611 + 0.973282i \(0.426254\pi\)
\(140\) −3243.10 −1.95780
\(141\) 1534.45 0.916480
\(142\) 2584.09 1.52713
\(143\) 0 0
\(144\) 1704.51 0.986404
\(145\) −1011.40 −0.579254
\(146\) −5105.34 −2.89398
\(147\) 747.911 0.419637
\(148\) 2009.13 1.11587
\(149\) −2636.72 −1.44972 −0.724862 0.688895i \(-0.758096\pi\)
−0.724862 + 0.688895i \(0.758096\pi\)
\(150\) −2308.96 −1.25684
\(151\) −3332.42 −1.79595 −0.897975 0.440046i \(-0.854962\pi\)
−0.897975 + 0.440046i \(0.854962\pi\)
\(152\) −148.692 −0.0793454
\(153\) 971.354 0.513263
\(154\) 1423.45 0.744839
\(155\) −3150.20 −1.63245
\(156\) 0 0
\(157\) −1625.26 −0.826179 −0.413089 0.910690i \(-0.635550\pi\)
−0.413089 + 0.910690i \(0.635550\pi\)
\(158\) −2000.05 −1.00706
\(159\) −1478.48 −0.737426
\(160\) −7881.36 −3.89423
\(161\) 404.698 0.198104
\(162\) 431.762 0.209398
\(163\) −1835.37 −0.881944 −0.440972 0.897521i \(-0.645366\pi\)
−0.440972 + 0.897521i \(0.645366\pi\)
\(164\) −627.605 −0.298828
\(165\) 1358.42 0.640927
\(166\) −3815.31 −1.78389
\(167\) 1945.00 0.901248 0.450624 0.892714i \(-0.351202\pi\)
0.450624 + 0.892714i \(0.351202\pi\)
\(168\) −1921.42 −0.882384
\(169\) 0 0
\(170\) −9442.44 −4.26001
\(171\) −20.2251 −0.00904474
\(172\) 4864.96 2.15668
\(173\) 2531.63 1.11258 0.556289 0.830989i \(-0.312225\pi\)
0.556289 + 0.830989i \(0.312225\pi\)
\(174\) −985.397 −0.429326
\(175\) 1397.65 0.603726
\(176\) 5224.91 2.23774
\(177\) −1452.54 −0.616832
\(178\) −5533.76 −2.33018
\(179\) 4263.01 1.78007 0.890035 0.455892i \(-0.150680\pi\)
0.890035 + 0.455892i \(0.150680\pi\)
\(180\) −3015.38 −1.24863
\(181\) 3944.61 1.61989 0.809946 0.586504i \(-0.199496\pi\)
0.809946 + 0.586504i \(0.199496\pi\)
\(182\) 0 0
\(183\) 1332.06 0.538082
\(184\) 2766.36 1.10836
\(185\) −1615.43 −0.641995
\(186\) −3069.22 −1.20992
\(187\) 2977.54 1.16438
\(188\) −10440.9 −4.05044
\(189\) −261.351 −0.100585
\(190\) 196.606 0.0750701
\(191\) −214.109 −0.0811119 −0.0405559 0.999177i \(-0.512913\pi\)
−0.0405559 + 0.999177i \(0.512913\pi\)
\(192\) −3133.42 −1.17779
\(193\) −1207.19 −0.450234 −0.225117 0.974332i \(-0.572276\pi\)
−0.225117 + 0.974332i \(0.572276\pi\)
\(194\) 349.480 0.129336
\(195\) 0 0
\(196\) −5089.06 −1.85461
\(197\) −927.631 −0.335487 −0.167744 0.985831i \(-0.553648\pi\)
−0.167744 + 0.985831i \(0.553648\pi\)
\(198\) 1323.50 0.475036
\(199\) 478.951 0.170613 0.0853064 0.996355i \(-0.472813\pi\)
0.0853064 + 0.996355i \(0.472813\pi\)
\(200\) 9553.77 3.37777
\(201\) −570.341 −0.200143
\(202\) 2834.69 0.987368
\(203\) 596.474 0.206228
\(204\) −6609.44 −2.26840
\(205\) 504.624 0.171924
\(206\) −3923.08 −1.32686
\(207\) 376.281 0.126345
\(208\) 0 0
\(209\) −61.9970 −0.0205188
\(210\) 2540.58 0.834840
\(211\) −1450.95 −0.473402 −0.236701 0.971583i \(-0.576066\pi\)
−0.236701 + 0.971583i \(0.576066\pi\)
\(212\) 10060.1 3.25910
\(213\) −1454.35 −0.467843
\(214\) −4175.11 −1.33367
\(215\) −3911.66 −1.24080
\(216\) −1786.50 −0.562759
\(217\) 1857.84 0.581191
\(218\) −2837.58 −0.881583
\(219\) 2873.34 0.886586
\(220\) −9243.19 −2.83262
\(221\) 0 0
\(222\) −1573.91 −0.475828
\(223\) −2059.79 −0.618536 −0.309268 0.950975i \(-0.600084\pi\)
−0.309268 + 0.950975i \(0.600084\pi\)
\(224\) 4648.06 1.38644
\(225\) 1299.51 0.385039
\(226\) −962.612 −0.283327
\(227\) 4482.46 1.31062 0.655311 0.755359i \(-0.272537\pi\)
0.655311 + 0.755359i \(0.272537\pi\)
\(228\) 137.619 0.0399738
\(229\) −1630.39 −0.470477 −0.235239 0.971938i \(-0.575587\pi\)
−0.235239 + 0.971938i \(0.575587\pi\)
\(230\) −3657.80 −1.04864
\(231\) −801.134 −0.228185
\(232\) 4077.27 1.15382
\(233\) −1903.69 −0.535258 −0.267629 0.963522i \(-0.586240\pi\)
−0.267629 + 0.963522i \(0.586240\pi\)
\(234\) 0 0
\(235\) 8395.00 2.33034
\(236\) 9883.58 2.72613
\(237\) 1125.65 0.308517
\(238\) 5568.72 1.51667
\(239\) 3763.79 1.01866 0.509328 0.860572i \(-0.329894\pi\)
0.509328 + 0.860572i \(0.329894\pi\)
\(240\) 9325.40 2.50813
\(241\) −3614.74 −0.966166 −0.483083 0.875575i \(-0.660483\pi\)
−0.483083 + 0.875575i \(0.660483\pi\)
\(242\) −3037.75 −0.806918
\(243\) −243.000 −0.0641500
\(244\) −9063.85 −2.37809
\(245\) 4091.84 1.06701
\(246\) 491.652 0.127425
\(247\) 0 0
\(248\) 12699.5 3.25169
\(249\) 2147.30 0.546503
\(250\) −1696.36 −0.429148
\(251\) −5729.77 −1.44088 −0.720438 0.693520i \(-0.756059\pi\)
−0.720438 + 0.693520i \(0.756059\pi\)
\(252\) 1778.33 0.444541
\(253\) 1153.43 0.286624
\(254\) 7631.18 1.88513
\(255\) 5314.30 1.30508
\(256\) 844.191 0.206101
\(257\) −5525.79 −1.34120 −0.670602 0.741818i \(-0.733964\pi\)
−0.670602 + 0.741818i \(0.733964\pi\)
\(258\) −3811.10 −0.919647
\(259\) 952.709 0.228565
\(260\) 0 0
\(261\) 554.591 0.131526
\(262\) 11019.6 2.59846
\(263\) 5223.21 1.22463 0.612313 0.790615i \(-0.290239\pi\)
0.612313 + 0.790615i \(0.290239\pi\)
\(264\) −5476.25 −1.27667
\(265\) −8088.78 −1.87506
\(266\) −115.949 −0.0267267
\(267\) 3114.46 0.713864
\(268\) 3880.81 0.884545
\(269\) 7203.88 1.63282 0.816410 0.577473i \(-0.195961\pi\)
0.816410 + 0.577473i \(0.195961\pi\)
\(270\) 2362.18 0.532436
\(271\) −8577.69 −1.92272 −0.961360 0.275293i \(-0.911225\pi\)
−0.961360 + 0.275293i \(0.911225\pi\)
\(272\) 20440.5 4.55656
\(273\) 0 0
\(274\) −2065.59 −0.455427
\(275\) 3983.44 0.873493
\(276\) −2560.35 −0.558388
\(277\) 7169.19 1.55507 0.777536 0.628838i \(-0.216469\pi\)
0.777536 + 0.628838i \(0.216469\pi\)
\(278\) 4011.48 0.865442
\(279\) 1727.39 0.370667
\(280\) −10512.1 −2.24364
\(281\) 849.157 0.180272 0.0901360 0.995929i \(-0.471270\pi\)
0.0901360 + 0.995929i \(0.471270\pi\)
\(282\) 8179.20 1.72718
\(283\) 1115.37 0.234283 0.117141 0.993115i \(-0.462627\pi\)
0.117141 + 0.993115i \(0.462627\pi\)
\(284\) 9895.95 2.06766
\(285\) −110.652 −0.0229981
\(286\) 0 0
\(287\) −297.604 −0.0612091
\(288\) 4321.69 0.884229
\(289\) 6735.49 1.37095
\(290\) −5391.14 −1.09165
\(291\) −196.691 −0.0396227
\(292\) −19551.2 −3.91832
\(293\) −1863.53 −0.371565 −0.185782 0.982591i \(-0.559482\pi\)
−0.185782 + 0.982591i \(0.559482\pi\)
\(294\) 3986.66 0.790839
\(295\) −7946.87 −1.56842
\(296\) 6512.36 1.27879
\(297\) −744.880 −0.145530
\(298\) −14054.8 −2.73212
\(299\) 0 0
\(300\) −8842.31 −1.70170
\(301\) 2306.91 0.441755
\(302\) −17763.1 −3.38461
\(303\) −1595.39 −0.302485
\(304\) −425.602 −0.0802959
\(305\) 7287.76 1.36818
\(306\) 5177.70 0.967285
\(307\) −6387.50 −1.18747 −0.593736 0.804660i \(-0.702348\pi\)
−0.593736 + 0.804660i \(0.702348\pi\)
\(308\) 5451.21 1.00848
\(309\) 2207.95 0.406492
\(310\) −16791.8 −3.07648
\(311\) −3492.59 −0.636806 −0.318403 0.947955i \(-0.603147\pi\)
−0.318403 + 0.947955i \(0.603147\pi\)
\(312\) 0 0
\(313\) −5912.01 −1.06762 −0.533812 0.845603i \(-0.679241\pi\)
−0.533812 + 0.845603i \(0.679241\pi\)
\(314\) −8663.29 −1.55700
\(315\) −1429.86 −0.255757
\(316\) −7659.31 −1.36351
\(317\) −1677.54 −0.297224 −0.148612 0.988896i \(-0.547481\pi\)
−0.148612 + 0.988896i \(0.547481\pi\)
\(318\) −7880.86 −1.38974
\(319\) 1700.02 0.298378
\(320\) −17143.0 −2.99476
\(321\) 2349.79 0.408575
\(322\) 2157.20 0.373342
\(323\) −242.540 −0.0417810
\(324\) 1653.46 0.283515
\(325\) 0 0
\(326\) −9783.22 −1.66209
\(327\) 1597.02 0.270077
\(328\) −2034.31 −0.342457
\(329\) −4950.98 −0.829655
\(330\) 7240.92 1.20788
\(331\) 2010.31 0.333827 0.166913 0.985972i \(-0.446620\pi\)
0.166913 + 0.985972i \(0.446620\pi\)
\(332\) −14611.0 −2.41531
\(333\) 885.812 0.145772
\(334\) 10367.6 1.69847
\(335\) −3120.35 −0.508905
\(336\) −5499.69 −0.892955
\(337\) 7139.24 1.15400 0.577002 0.816743i \(-0.304222\pi\)
0.577002 + 0.816743i \(0.304222\pi\)
\(338\) 0 0
\(339\) 541.768 0.0867988
\(340\) −36160.5 −5.76787
\(341\) 5295.05 0.840889
\(342\) −107.808 −0.0170455
\(343\) −5733.31 −0.902536
\(344\) 15769.2 2.47156
\(345\) 2058.64 0.321257
\(346\) 13494.6 2.09674
\(347\) 1.13990 0.000176349 0 8.81743e−5 1.00000i \(-0.499972\pi\)
8.81743e−5 1.00000i \(0.499972\pi\)
\(348\) −3773.64 −0.581289
\(349\) 12199.1 1.87107 0.935535 0.353235i \(-0.114918\pi\)
0.935535 + 0.353235i \(0.114918\pi\)
\(350\) 7450.00 1.13777
\(351\) 0 0
\(352\) 13247.5 2.00595
\(353\) −10892.3 −1.64232 −0.821160 0.570698i \(-0.806673\pi\)
−0.821160 + 0.570698i \(0.806673\pi\)
\(354\) −7742.59 −1.16247
\(355\) −7956.81 −1.18959
\(356\) −21191.9 −3.15497
\(357\) −3134.13 −0.464638
\(358\) 22723.5 3.35468
\(359\) −3525.78 −0.518339 −0.259169 0.965832i \(-0.583449\pi\)
−0.259169 + 0.965832i \(0.583449\pi\)
\(360\) −9773.99 −1.43093
\(361\) −6853.95 −0.999264
\(362\) 21026.3 3.05281
\(363\) 1709.68 0.247204
\(364\) 0 0
\(365\) 15720.1 2.25433
\(366\) 7100.42 1.01406
\(367\) 2383.75 0.339049 0.169525 0.985526i \(-0.445777\pi\)
0.169525 + 0.985526i \(0.445777\pi\)
\(368\) 7918.19 1.12164
\(369\) −276.707 −0.0390374
\(370\) −8610.90 −1.20989
\(371\) 4770.39 0.667564
\(372\) −11753.8 −1.63818
\(373\) −13282.2 −1.84377 −0.921885 0.387463i \(-0.873352\pi\)
−0.921885 + 0.387463i \(0.873352\pi\)
\(374\) 15871.5 2.19437
\(375\) 954.727 0.131472
\(376\) −33843.1 −4.64181
\(377\) 0 0
\(378\) −1393.11 −0.189560
\(379\) −4436.73 −0.601318 −0.300659 0.953732i \(-0.597207\pi\)
−0.300659 + 0.953732i \(0.597207\pi\)
\(380\) 752.917 0.101642
\(381\) −4294.90 −0.577519
\(382\) −1141.28 −0.152862
\(383\) −810.412 −0.108120 −0.0540602 0.998538i \(-0.517216\pi\)
−0.0540602 + 0.998538i \(0.517216\pi\)
\(384\) −5177.83 −0.688100
\(385\) −4383.03 −0.580207
\(386\) −6434.78 −0.848501
\(387\) 2144.93 0.281739
\(388\) 1338.35 0.175115
\(389\) 3463.79 0.451469 0.225734 0.974189i \(-0.427522\pi\)
0.225734 + 0.974189i \(0.427522\pi\)
\(390\) 0 0
\(391\) 4512.37 0.583633
\(392\) −16495.6 −2.12539
\(393\) −6201.96 −0.796050
\(394\) −4944.64 −0.632252
\(395\) 6158.45 0.784469
\(396\) 5068.44 0.643178
\(397\) 425.405 0.0537796 0.0268898 0.999638i \(-0.491440\pi\)
0.0268898 + 0.999638i \(0.491440\pi\)
\(398\) 2553.00 0.321533
\(399\) 65.2575 0.00818787
\(400\) 27345.9 3.41823
\(401\) −1186.85 −0.147801 −0.0739007 0.997266i \(-0.523545\pi\)
−0.0739007 + 0.997266i \(0.523545\pi\)
\(402\) −3040.14 −0.377185
\(403\) 0 0
\(404\) 10855.6 1.33685
\(405\) −1329.46 −0.163115
\(406\) 3179.44 0.388653
\(407\) 2715.33 0.330697
\(408\) −21423.7 −2.59959
\(409\) 8007.42 0.968071 0.484036 0.875048i \(-0.339170\pi\)
0.484036 + 0.875048i \(0.339170\pi\)
\(410\) 2689.84 0.324005
\(411\) 1162.54 0.139522
\(412\) −15023.7 −1.79652
\(413\) 4686.70 0.558395
\(414\) 2005.73 0.238106
\(415\) 11747.9 1.38960
\(416\) 0 0
\(417\) −2257.70 −0.265132
\(418\) −330.468 −0.0386692
\(419\) 6832.46 0.796629 0.398314 0.917249i \(-0.369595\pi\)
0.398314 + 0.917249i \(0.369595\pi\)
\(420\) 9729.30 1.13034
\(421\) 10739.6 1.24326 0.621632 0.783309i \(-0.286470\pi\)
0.621632 + 0.783309i \(0.286470\pi\)
\(422\) −7734.16 −0.892163
\(423\) −4603.34 −0.529130
\(424\) 32608.6 3.73494
\(425\) 15583.7 1.77864
\(426\) −7752.28 −0.881688
\(427\) −4297.99 −0.487106
\(428\) −15988.9 −1.80573
\(429\) 0 0
\(430\) −20850.7 −2.33839
\(431\) 5214.45 0.582763 0.291382 0.956607i \(-0.405885\pi\)
0.291382 + 0.956607i \(0.405885\pi\)
\(432\) −5113.52 −0.569501
\(433\) 8642.24 0.959168 0.479584 0.877496i \(-0.340788\pi\)
0.479584 + 0.877496i \(0.340788\pi\)
\(434\) 9903.02 1.09530
\(435\) 3034.19 0.334432
\(436\) −10866.7 −1.19362
\(437\) −93.9545 −0.0102848
\(438\) 15316.0 1.67084
\(439\) −13026.2 −1.41619 −0.708097 0.706116i \(-0.750446\pi\)
−0.708097 + 0.706116i \(0.750446\pi\)
\(440\) −29960.7 −3.24619
\(441\) −2243.73 −0.242278
\(442\) 0 0
\(443\) −11533.0 −1.23690 −0.618450 0.785824i \(-0.712239\pi\)
−0.618450 + 0.785824i \(0.712239\pi\)
\(444\) −6027.39 −0.644250
\(445\) 17039.3 1.81515
\(446\) −10979.5 −1.16568
\(447\) 7910.17 0.836998
\(448\) 10110.2 1.06621
\(449\) 9882.75 1.03874 0.519372 0.854548i \(-0.326166\pi\)
0.519372 + 0.854548i \(0.326166\pi\)
\(450\) 6926.88 0.725636
\(451\) −848.204 −0.0885596
\(452\) −3686.38 −0.383613
\(453\) 9997.26 1.03689
\(454\) 23893.3 2.46997
\(455\) 0 0
\(456\) 446.075 0.0458101
\(457\) −15628.1 −1.59967 −0.799836 0.600218i \(-0.795080\pi\)
−0.799836 + 0.600218i \(0.795080\pi\)
\(458\) −8690.63 −0.886652
\(459\) −2914.06 −0.296333
\(460\) −14007.8 −1.41982
\(461\) −7747.46 −0.782723 −0.391361 0.920237i \(-0.627996\pi\)
−0.391361 + 0.920237i \(0.627996\pi\)
\(462\) −4270.36 −0.430033
\(463\) −333.422 −0.0334675 −0.0167337 0.999860i \(-0.505327\pi\)
−0.0167337 + 0.999860i \(0.505327\pi\)
\(464\) 11670.4 1.16764
\(465\) 9450.59 0.942496
\(466\) −10147.4 −1.00874
\(467\) 8198.33 0.812363 0.406182 0.913792i \(-0.366860\pi\)
0.406182 + 0.913792i \(0.366860\pi\)
\(468\) 0 0
\(469\) 1840.24 0.181182
\(470\) 44748.7 4.39171
\(471\) 4875.79 0.476995
\(472\) 32036.5 3.12415
\(473\) 6574.96 0.639148
\(474\) 6000.14 0.581425
\(475\) −324.477 −0.0313432
\(476\) 21325.8 2.05350
\(477\) 4435.43 0.425753
\(478\) 20062.5 1.91974
\(479\) −6435.88 −0.613910 −0.306955 0.951724i \(-0.599310\pi\)
−0.306955 + 0.951724i \(0.599310\pi\)
\(480\) 23644.1 2.24833
\(481\) 0 0
\(482\) −19268.0 −1.82082
\(483\) −1214.09 −0.114375
\(484\) −11633.3 −1.09253
\(485\) −1076.10 −0.100749
\(486\) −1295.29 −0.120896
\(487\) 8095.37 0.753257 0.376629 0.926364i \(-0.377083\pi\)
0.376629 + 0.926364i \(0.377083\pi\)
\(488\) −29379.4 −2.72529
\(489\) 5506.10 0.509191
\(490\) 21811.1 2.01087
\(491\) 5116.46 0.470270 0.235135 0.971963i \(-0.424447\pi\)
0.235135 + 0.971963i \(0.424447\pi\)
\(492\) 1882.82 0.172528
\(493\) 6650.67 0.607568
\(494\) 0 0
\(495\) −4075.26 −0.370039
\(496\) 36349.9 3.29064
\(497\) 4692.56 0.423521
\(498\) 11445.9 1.02993
\(499\) −18050.7 −1.61936 −0.809682 0.586870i \(-0.800360\pi\)
−0.809682 + 0.586870i \(0.800360\pi\)
\(500\) −6496.31 −0.581047
\(501\) −5834.99 −0.520336
\(502\) −30541.9 −2.71544
\(503\) 10531.1 0.933512 0.466756 0.884386i \(-0.345423\pi\)
0.466756 + 0.884386i \(0.345423\pi\)
\(504\) 5764.25 0.509445
\(505\) −8728.45 −0.769131
\(506\) 6148.25 0.540165
\(507\) 0 0
\(508\) 29224.1 2.55238
\(509\) −1963.31 −0.170967 −0.0854834 0.996340i \(-0.527243\pi\)
−0.0854834 + 0.996340i \(0.527243\pi\)
\(510\) 28327.3 2.45952
\(511\) −9271.00 −0.802593
\(512\) −9307.69 −0.803409
\(513\) 60.6753 0.00522198
\(514\) −29454.6 −2.52760
\(515\) 12079.8 1.03359
\(516\) −14594.9 −1.24516
\(517\) −14110.9 −1.20038
\(518\) 5078.31 0.430750
\(519\) −7594.89 −0.642348
\(520\) 0 0
\(521\) −7044.93 −0.592407 −0.296203 0.955125i \(-0.595721\pi\)
−0.296203 + 0.955125i \(0.595721\pi\)
\(522\) 2956.19 0.247872
\(523\) −3213.29 −0.268657 −0.134328 0.990937i \(-0.542888\pi\)
−0.134328 + 0.990937i \(0.542888\pi\)
\(524\) 42200.4 3.51819
\(525\) −4192.94 −0.348561
\(526\) 27841.7 2.30790
\(527\) 20714.9 1.71225
\(528\) −15674.7 −1.29196
\(529\) −10419.0 −0.856333
\(530\) −43116.4 −3.53369
\(531\) 4357.61 0.356128
\(532\) −444.036 −0.0361868
\(533\) 0 0
\(534\) 16601.3 1.34533
\(535\) 12855.8 1.03889
\(536\) 12579.2 1.01369
\(537\) −12789.0 −1.02772
\(538\) 38399.5 3.07718
\(539\) −6877.83 −0.549627
\(540\) 9046.13 0.720895
\(541\) 11251.4 0.894150 0.447075 0.894497i \(-0.352466\pi\)
0.447075 + 0.894497i \(0.352466\pi\)
\(542\) −45722.4 −3.62352
\(543\) −11833.8 −0.935245
\(544\) 51825.8 4.08458
\(545\) 8737.33 0.686727
\(546\) 0 0
\(547\) 1533.54 0.119871 0.0599353 0.998202i \(-0.480911\pi\)
0.0599353 + 0.998202i \(0.480911\pi\)
\(548\) −7910.31 −0.616628
\(549\) −3996.19 −0.310662
\(550\) 21233.3 1.64617
\(551\) −138.477 −0.0107066
\(552\) −8299.08 −0.639914
\(553\) −3631.97 −0.279289
\(554\) 38214.6 2.93066
\(555\) 4846.30 0.370656
\(556\) 15362.2 1.17177
\(557\) 16845.7 1.28146 0.640731 0.767766i \(-0.278631\pi\)
0.640731 + 0.767766i \(0.278631\pi\)
\(558\) 9207.65 0.698550
\(559\) 0 0
\(560\) −30089.0 −2.27052
\(561\) −8932.62 −0.672256
\(562\) 4526.34 0.339737
\(563\) −20820.1 −1.55855 −0.779273 0.626685i \(-0.784411\pi\)
−0.779273 + 0.626685i \(0.784411\pi\)
\(564\) 31322.8 2.33852
\(565\) 2964.03 0.220704
\(566\) 5945.37 0.441524
\(567\) 784.054 0.0580726
\(568\) 32076.6 2.36955
\(569\) 23636.6 1.74147 0.870735 0.491752i \(-0.163643\pi\)
0.870735 + 0.491752i \(0.163643\pi\)
\(570\) −589.819 −0.0433417
\(571\) −26955.1 −1.97554 −0.987771 0.155913i \(-0.950168\pi\)
−0.987771 + 0.155913i \(0.950168\pi\)
\(572\) 0 0
\(573\) 642.326 0.0468300
\(574\) −1586.35 −0.115353
\(575\) 6036.78 0.437828
\(576\) 9400.25 0.679995
\(577\) 23499.8 1.69551 0.847755 0.530388i \(-0.177954\pi\)
0.847755 + 0.530388i \(0.177954\pi\)
\(578\) 35902.8 2.58367
\(579\) 3621.56 0.259943
\(580\) −20645.7 −1.47805
\(581\) −6928.38 −0.494729
\(582\) −1048.44 −0.0746721
\(583\) 13596.1 0.965857
\(584\) −63373.1 −4.49040
\(585\) 0 0
\(586\) −9933.33 −0.700243
\(587\) 4637.50 0.326082 0.163041 0.986619i \(-0.447870\pi\)
0.163041 + 0.986619i \(0.447870\pi\)
\(588\) 15267.2 1.07076
\(589\) −431.316 −0.0301733
\(590\) −42359.9 −2.95582
\(591\) 2782.89 0.193694
\(592\) 18640.4 1.29411
\(593\) 12633.5 0.874869 0.437434 0.899250i \(-0.355887\pi\)
0.437434 + 0.899250i \(0.355887\pi\)
\(594\) −3970.51 −0.274262
\(595\) −17146.9 −1.18144
\(596\) −53823.7 −3.69917
\(597\) −1436.85 −0.0985033
\(598\) 0 0
\(599\) −18757.1 −1.27946 −0.639730 0.768600i \(-0.720954\pi\)
−0.639730 + 0.768600i \(0.720954\pi\)
\(600\) −28661.3 −1.95016
\(601\) −3632.98 −0.246576 −0.123288 0.992371i \(-0.539344\pi\)
−0.123288 + 0.992371i \(0.539344\pi\)
\(602\) 12296.8 0.832522
\(603\) 1711.02 0.115553
\(604\) −68025.0 −4.58261
\(605\) 9353.71 0.628566
\(606\) −8504.08 −0.570057
\(607\) −12700.0 −0.849219 −0.424610 0.905377i \(-0.639589\pi\)
−0.424610 + 0.905377i \(0.639589\pi\)
\(608\) −1079.09 −0.0719786
\(609\) −1789.42 −0.119066
\(610\) 38846.6 2.57845
\(611\) 0 0
\(612\) 19828.3 1.30966
\(613\) 21640.1 1.42584 0.712918 0.701248i \(-0.247373\pi\)
0.712918 + 0.701248i \(0.247373\pi\)
\(614\) −34047.9 −2.23788
\(615\) −1513.87 −0.0992605
\(616\) 17669.5 1.15572
\(617\) −16541.7 −1.07933 −0.539663 0.841881i \(-0.681448\pi\)
−0.539663 + 0.841881i \(0.681448\pi\)
\(618\) 11769.2 0.766066
\(619\) −21138.9 −1.37261 −0.686303 0.727316i \(-0.740767\pi\)
−0.686303 + 0.727316i \(0.740767\pi\)
\(620\) −64305.2 −4.16542
\(621\) −1128.84 −0.0729451
\(622\) −18616.9 −1.20011
\(623\) −10049.0 −0.646235
\(624\) 0 0
\(625\) −12825.4 −0.820823
\(626\) −31513.3 −2.01202
\(627\) 185.991 0.0118465
\(628\) −33176.6 −2.10811
\(629\) 10622.7 0.673377
\(630\) −7621.73 −0.481995
\(631\) 5489.80 0.346348 0.173174 0.984891i \(-0.444598\pi\)
0.173174 + 0.984891i \(0.444598\pi\)
\(632\) −24826.8 −1.56259
\(633\) 4352.86 0.273319
\(634\) −8941.94 −0.560141
\(635\) −23497.5 −1.46846
\(636\) −30180.3 −1.88164
\(637\) 0 0
\(638\) 9061.76 0.562318
\(639\) 4363.06 0.270110
\(640\) −28328.1 −1.74963
\(641\) 4297.04 0.264778 0.132389 0.991198i \(-0.457735\pi\)
0.132389 + 0.991198i \(0.457735\pi\)
\(642\) 12525.3 0.769993
\(643\) 25696.9 1.57603 0.788016 0.615655i \(-0.211108\pi\)
0.788016 + 0.615655i \(0.211108\pi\)
\(644\) 8261.14 0.505488
\(645\) 11735.0 0.716378
\(646\) −1292.83 −0.0787396
\(647\) 2174.98 0.132160 0.0660798 0.997814i \(-0.478951\pi\)
0.0660798 + 0.997814i \(0.478951\pi\)
\(648\) 5359.50 0.324909
\(649\) 13357.6 0.807907
\(650\) 0 0
\(651\) −5573.52 −0.335551
\(652\) −37465.5 −2.25040
\(653\) −15454.5 −0.926160 −0.463080 0.886316i \(-0.653256\pi\)
−0.463080 + 0.886316i \(0.653256\pi\)
\(654\) 8512.73 0.508982
\(655\) −33931.1 −2.02412
\(656\) −5822.82 −0.346560
\(657\) −8620.02 −0.511870
\(658\) −26390.7 −1.56355
\(659\) −3148.77 −0.186129 −0.0930643 0.995660i \(-0.529666\pi\)
−0.0930643 + 0.995660i \(0.529666\pi\)
\(660\) 27729.6 1.63541
\(661\) −2099.70 −0.123553 −0.0617767 0.998090i \(-0.519677\pi\)
−0.0617767 + 0.998090i \(0.519677\pi\)
\(662\) 10715.7 0.629122
\(663\) 0 0
\(664\) −47359.8 −2.76795
\(665\) 357.026 0.0208193
\(666\) 4721.73 0.274720
\(667\) 2576.32 0.149559
\(668\) 39703.4 2.29966
\(669\) 6179.36 0.357112
\(670\) −16632.7 −0.959071
\(671\) −12249.7 −0.704763
\(672\) −13944.2 −0.800460
\(673\) 30970.8 1.77390 0.886950 0.461865i \(-0.152819\pi\)
0.886950 + 0.461865i \(0.152819\pi\)
\(674\) 38055.0 2.17481
\(675\) −3898.52 −0.222302
\(676\) 0 0
\(677\) 14640.6 0.831141 0.415570 0.909561i \(-0.363582\pi\)
0.415570 + 0.909561i \(0.363582\pi\)
\(678\) 2887.83 0.163579
\(679\) 634.635 0.0358690
\(680\) −117210. −6.60999
\(681\) −13447.4 −0.756688
\(682\) 28224.7 1.58472
\(683\) −6685.83 −0.374563 −0.187281 0.982306i \(-0.559968\pi\)
−0.187281 + 0.982306i \(0.559968\pi\)
\(684\) −412.856 −0.0230789
\(685\) 6360.27 0.354764
\(686\) −30560.8 −1.70090
\(687\) 4891.18 0.271630
\(688\) 45136.3 2.50117
\(689\) 0 0
\(690\) 10973.4 0.605434
\(691\) −30194.1 −1.66228 −0.831141 0.556062i \(-0.812312\pi\)
−0.831141 + 0.556062i \(0.812312\pi\)
\(692\) 51678.4 2.83890
\(693\) 2403.40 0.131743
\(694\) 6.07611 0.000332343 0
\(695\) −12352.0 −0.674154
\(696\) −12231.8 −0.666158
\(697\) −3318.28 −0.180328
\(698\) 65026.0 3.52618
\(699\) 5711.08 0.309031
\(700\) 28530.3 1.54049
\(701\) −30300.9 −1.63260 −0.816298 0.577631i \(-0.803977\pi\)
−0.816298 + 0.577631i \(0.803977\pi\)
\(702\) 0 0
\(703\) −221.181 −0.0118663
\(704\) 28815.1 1.54263
\(705\) −25185.0 −1.34542
\(706\) −58060.3 −3.09508
\(707\) 5147.64 0.273829
\(708\) −29650.8 −1.57393
\(709\) 26123.2 1.38375 0.691875 0.722017i \(-0.256785\pi\)
0.691875 + 0.722017i \(0.256785\pi\)
\(710\) −42412.9 −2.24187
\(711\) −3376.94 −0.178123
\(712\) −68691.1 −3.61560
\(713\) 8024.48 0.421486
\(714\) −16706.2 −0.875647
\(715\) 0 0
\(716\) 87021.3 4.54209
\(717\) −11291.4 −0.588122
\(718\) −18793.8 −0.976850
\(719\) 19325.7 1.00240 0.501200 0.865331i \(-0.332892\pi\)
0.501200 + 0.865331i \(0.332892\pi\)
\(720\) −27976.2 −1.44807
\(721\) −7124.09 −0.367982
\(722\) −36534.2 −1.88319
\(723\) 10844.2 0.557816
\(724\) 80521.7 4.13338
\(725\) 8897.47 0.455784
\(726\) 9113.26 0.465875
\(727\) 26065.8 1.32975 0.664875 0.746954i \(-0.268485\pi\)
0.664875 + 0.746954i \(0.268485\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 83794.4 4.24845
\(731\) 25722.0 1.30145
\(732\) 27191.5 1.37299
\(733\) 1055.45 0.0531843 0.0265921 0.999646i \(-0.491534\pi\)
0.0265921 + 0.999646i \(0.491534\pi\)
\(734\) 12706.4 0.638965
\(735\) −12275.5 −0.616041
\(736\) 20076.2 1.00546
\(737\) 5244.89 0.262141
\(738\) −1474.96 −0.0735690
\(739\) 9410.40 0.468426 0.234213 0.972185i \(-0.424749\pi\)
0.234213 + 0.972185i \(0.424749\pi\)
\(740\) −32976.0 −1.63814
\(741\) 0 0
\(742\) 25428.1 1.25808
\(743\) −7523.70 −0.371491 −0.185746 0.982598i \(-0.559470\pi\)
−0.185746 + 0.982598i \(0.559470\pi\)
\(744\) −38098.5 −1.87736
\(745\) 43276.8 2.12824
\(746\) −70799.4 −3.47473
\(747\) −6441.89 −0.315524
\(748\) 60780.8 2.97108
\(749\) −7581.76 −0.369868
\(750\) 5089.07 0.247769
\(751\) −12984.1 −0.630886 −0.315443 0.948945i \(-0.602153\pi\)
−0.315443 + 0.948945i \(0.602153\pi\)
\(752\) −96869.3 −4.69742
\(753\) 17189.3 0.831890
\(754\) 0 0
\(755\) 54695.3 2.63651
\(756\) −5334.99 −0.256656
\(757\) −27934.6 −1.34122 −0.670609 0.741811i \(-0.733967\pi\)
−0.670609 + 0.741811i \(0.733967\pi\)
\(758\) −23649.5 −1.13323
\(759\) −3460.30 −0.165482
\(760\) 2440.49 0.116482
\(761\) −15519.3 −0.739255 −0.369627 0.929180i \(-0.620515\pi\)
−0.369627 + 0.929180i \(0.620515\pi\)
\(762\) −22893.5 −1.08838
\(763\) −5152.88 −0.244491
\(764\) −4370.62 −0.206968
\(765\) −15942.9 −0.753487
\(766\) −4319.82 −0.203761
\(767\) 0 0
\(768\) −2532.57 −0.118993
\(769\) 12885.2 0.604228 0.302114 0.953272i \(-0.402308\pi\)
0.302114 + 0.953272i \(0.402308\pi\)
\(770\) −23363.3 −1.09345
\(771\) 16577.4 0.774344
\(772\) −24642.4 −1.14883
\(773\) 5892.04 0.274155 0.137078 0.990560i \(-0.456229\pi\)
0.137078 + 0.990560i \(0.456229\pi\)
\(774\) 11433.3 0.530958
\(775\) 27713.0 1.28449
\(776\) 4338.12 0.200682
\(777\) −2858.13 −0.131962
\(778\) 18463.4 0.850828
\(779\) 69.0916 0.00317775
\(780\) 0 0
\(781\) 13374.3 0.612766
\(782\) 24052.7 1.09990
\(783\) −1663.77 −0.0759367
\(784\) −47215.5 −2.15085
\(785\) 26675.6 1.21286
\(786\) −33058.9 −1.50022
\(787\) −21020.4 −0.952091 −0.476045 0.879421i \(-0.657930\pi\)
−0.476045 + 0.879421i \(0.657930\pi\)
\(788\) −18935.8 −0.856042
\(789\) −15669.6 −0.707038
\(790\) 32827.0 1.47839
\(791\) −1748.05 −0.0785757
\(792\) 16428.7 0.737084
\(793\) 0 0
\(794\) 2267.58 0.101352
\(795\) 24266.4 1.08256
\(796\) 9776.87 0.435342
\(797\) 31355.6 1.39356 0.696782 0.717283i \(-0.254614\pi\)
0.696782 + 0.717283i \(0.254614\pi\)
\(798\) 347.848 0.0154307
\(799\) −55203.3 −2.44425
\(800\) 69334.0 3.06416
\(801\) −9343.37 −0.412150
\(802\) −6326.37 −0.278543
\(803\) −26423.4 −1.16122
\(804\) −11642.4 −0.510692
\(805\) −6642.34 −0.290822
\(806\) 0 0
\(807\) −21611.7 −0.942709
\(808\) 35187.3 1.53204
\(809\) −18132.5 −0.788017 −0.394009 0.919107i \(-0.628912\pi\)
−0.394009 + 0.919107i \(0.628912\pi\)
\(810\) −7086.55 −0.307402
\(811\) −24755.3 −1.07186 −0.535928 0.844263i \(-0.680038\pi\)
−0.535928 + 0.844263i \(0.680038\pi\)
\(812\) 12175.9 0.526219
\(813\) 25733.1 1.11008
\(814\) 14473.8 0.623225
\(815\) 30124.0 1.29472
\(816\) −61321.4 −2.63073
\(817\) −535.572 −0.0229343
\(818\) 42682.7 1.82441
\(819\) 0 0
\(820\) 10300.9 0.438688
\(821\) 4082.65 0.173551 0.0867755 0.996228i \(-0.472344\pi\)
0.0867755 + 0.996228i \(0.472344\pi\)
\(822\) 6196.77 0.262941
\(823\) −34327.0 −1.45391 −0.726954 0.686687i \(-0.759065\pi\)
−0.726954 + 0.686687i \(0.759065\pi\)
\(824\) −48697.6 −2.05881
\(825\) −11950.3 −0.504312
\(826\) 24981.9 1.05234
\(827\) 3228.87 0.135767 0.0678833 0.997693i \(-0.478375\pi\)
0.0678833 + 0.997693i \(0.478375\pi\)
\(828\) 7681.06 0.322386
\(829\) −10452.4 −0.437908 −0.218954 0.975735i \(-0.570265\pi\)
−0.218954 + 0.975735i \(0.570265\pi\)
\(830\) 62621.0 2.61880
\(831\) −21507.6 −0.897821
\(832\) 0 0
\(833\) −26906.9 −1.11917
\(834\) −12034.5 −0.499663
\(835\) −31923.4 −1.32306
\(836\) −1265.55 −0.0523564
\(837\) −5182.16 −0.214004
\(838\) 36419.7 1.50131
\(839\) −28289.0 −1.16406 −0.582028 0.813169i \(-0.697741\pi\)
−0.582028 + 0.813169i \(0.697741\pi\)
\(840\) 31536.4 1.29537
\(841\) −20591.8 −0.844308
\(842\) 57246.1 2.34303
\(843\) −2547.47 −0.104080
\(844\) −29618.5 −1.20795
\(845\) 0 0
\(846\) −24537.6 −0.997187
\(847\) −5516.39 −0.223784
\(848\) 93335.9 3.77968
\(849\) −3346.12 −0.135263
\(850\) 83067.2 3.35198
\(851\) 4114.99 0.165758
\(852\) −29687.8 −1.19377
\(853\) −26631.8 −1.06900 −0.534498 0.845170i \(-0.679499\pi\)
−0.534498 + 0.845170i \(0.679499\pi\)
\(854\) −22910.0 −0.917989
\(855\) 331.956 0.0132780
\(856\) −51826.0 −2.06936
\(857\) 11796.7 0.470209 0.235104 0.971970i \(-0.424457\pi\)
0.235104 + 0.971970i \(0.424457\pi\)
\(858\) 0 0
\(859\) −22672.8 −0.900567 −0.450283 0.892886i \(-0.648677\pi\)
−0.450283 + 0.892886i \(0.648677\pi\)
\(860\) −79849.0 −3.16608
\(861\) 892.812 0.0353391
\(862\) 27795.0 1.09826
\(863\) 21421.1 0.844940 0.422470 0.906377i \(-0.361163\pi\)
0.422470 + 0.906377i \(0.361163\pi\)
\(864\) −12965.1 −0.510510
\(865\) −41551.9 −1.63330
\(866\) 46066.6 1.80763
\(867\) −20206.5 −0.791520
\(868\) 37924.3 1.48299
\(869\) −10351.5 −0.404086
\(870\) 16173.4 0.630264
\(871\) 0 0
\(872\) −35223.1 −1.36790
\(873\) 590.072 0.0228762
\(874\) −500.814 −0.0193825
\(875\) −3080.48 −0.119016
\(876\) 58653.7 2.26224
\(877\) −5155.20 −0.198493 −0.0992466 0.995063i \(-0.531643\pi\)
−0.0992466 + 0.995063i \(0.531643\pi\)
\(878\) −69435.0 −2.66893
\(879\) 5590.58 0.214523
\(880\) −85756.9 −3.28507
\(881\) 23692.2 0.906027 0.453013 0.891504i \(-0.350349\pi\)
0.453013 + 0.891504i \(0.350349\pi\)
\(882\) −11960.0 −0.456591
\(883\) −14591.5 −0.556108 −0.278054 0.960565i \(-0.589689\pi\)
−0.278054 + 0.960565i \(0.589689\pi\)
\(884\) 0 0
\(885\) 23840.6 0.905529
\(886\) −61475.2 −2.33104
\(887\) −9722.26 −0.368029 −0.184014 0.982924i \(-0.558909\pi\)
−0.184014 + 0.982924i \(0.558909\pi\)
\(888\) −19537.1 −0.738312
\(889\) 13857.8 0.522806
\(890\) 90826.1 3.42078
\(891\) 2234.64 0.0840217
\(892\) −42046.6 −1.57828
\(893\) 1149.42 0.0430726
\(894\) 42164.3 1.57739
\(895\) −69969.2 −2.61320
\(896\) 16706.6 0.622911
\(897\) 0 0
\(898\) 52678.9 1.95759
\(899\) 11827.1 0.438771
\(900\) 26526.9 0.982479
\(901\) 53189.7 1.96671
\(902\) −4521.26 −0.166898
\(903\) −6920.74 −0.255048
\(904\) −11949.0 −0.439621
\(905\) −64743.2 −2.37805
\(906\) 53289.3 1.95411
\(907\) 11799.0 0.431951 0.215975 0.976399i \(-0.430707\pi\)
0.215975 + 0.976399i \(0.430707\pi\)
\(908\) 91500.9 3.34423
\(909\) 4786.18 0.174640
\(910\) 0 0
\(911\) −43012.4 −1.56429 −0.782143 0.623099i \(-0.785873\pi\)
−0.782143 + 0.623099i \(0.785873\pi\)
\(912\) 1276.81 0.0463589
\(913\) −19746.6 −0.715793
\(914\) −83303.8 −3.01471
\(915\) −21863.3 −0.789921
\(916\) −33281.3 −1.20049
\(917\) 20011.0 0.720635
\(918\) −15533.1 −0.558462
\(919\) −4951.41 −0.177728 −0.0888639 0.996044i \(-0.528324\pi\)
−0.0888639 + 0.996044i \(0.528324\pi\)
\(920\) −45404.5 −1.62711
\(921\) 19162.5 0.685587
\(922\) −41297.0 −1.47510
\(923\) 0 0
\(924\) −16353.6 −0.582245
\(925\) 14211.3 0.505152
\(926\) −1777.27 −0.0630721
\(927\) −6623.85 −0.234688
\(928\) 29589.8 1.04669
\(929\) −8934.86 −0.315547 −0.157774 0.987475i \(-0.550432\pi\)
−0.157774 + 0.987475i \(0.550432\pi\)
\(930\) 50375.4 1.77621
\(931\) 560.243 0.0197221
\(932\) −38860.2 −1.36578
\(933\) 10477.8 0.367660
\(934\) 43700.3 1.53096
\(935\) −48870.6 −1.70935
\(936\) 0 0
\(937\) −13182.8 −0.459620 −0.229810 0.973235i \(-0.573811\pi\)
−0.229810 + 0.973235i \(0.573811\pi\)
\(938\) 9809.20 0.341452
\(939\) 17736.0 0.616393
\(940\) 171368. 5.94618
\(941\) −21693.7 −0.751536 −0.375768 0.926714i \(-0.622621\pi\)
−0.375768 + 0.926714i \(0.622621\pi\)
\(942\) 25989.9 0.898934
\(943\) −1285.43 −0.0443895
\(944\) 91698.4 3.16158
\(945\) 4289.59 0.147662
\(946\) 35047.1 1.20452
\(947\) −49790.0 −1.70851 −0.854254 0.519856i \(-0.825986\pi\)
−0.854254 + 0.519856i \(0.825986\pi\)
\(948\) 22977.9 0.787224
\(949\) 0 0
\(950\) −1729.59 −0.0590687
\(951\) 5032.61 0.171602
\(952\) 69125.0 2.35331
\(953\) 4217.93 0.143371 0.0716853 0.997427i \(-0.477162\pi\)
0.0716853 + 0.997427i \(0.477162\pi\)
\(954\) 23642.6 0.802365
\(955\) 3514.19 0.119075
\(956\) 76830.5 2.59924
\(957\) −5100.05 −0.172269
\(958\) −34305.8 −1.15696
\(959\) −3750.99 −0.126304
\(960\) 51429.0 1.72903
\(961\) 7046.87 0.236544
\(962\) 0 0
\(963\) −7049.38 −0.235891
\(964\) −73788.1 −2.46530
\(965\) 19813.7 0.660958
\(966\) −6471.60 −0.215549
\(967\) −40927.9 −1.36107 −0.680534 0.732717i \(-0.738252\pi\)
−0.680534 + 0.732717i \(0.738252\pi\)
\(968\) −37707.9 −1.25204
\(969\) 727.619 0.0241223
\(970\) −5736.04 −0.189869
\(971\) −17114.8 −0.565645 −0.282822 0.959172i \(-0.591271\pi\)
−0.282822 + 0.959172i \(0.591271\pi\)
\(972\) −4960.38 −0.163688
\(973\) 7284.62 0.240015
\(974\) 43151.5 1.41957
\(975\) 0 0
\(976\) −84093.0 −2.75794
\(977\) −118.470 −0.00387940 −0.00193970 0.999998i \(-0.500617\pi\)
−0.00193970 + 0.999998i \(0.500617\pi\)
\(978\) 29349.7 0.959610
\(979\) −28640.7 −0.934996
\(980\) 83527.2 2.72263
\(981\) −4791.05 −0.155929
\(982\) 27272.8 0.886261
\(983\) 26002.8 0.843705 0.421852 0.906665i \(-0.361380\pi\)
0.421852 + 0.906665i \(0.361380\pi\)
\(984\) 6102.93 0.197718
\(985\) 15225.3 0.492506
\(986\) 35450.7 1.14501
\(987\) 14853.0 0.479002
\(988\) 0 0
\(989\) 9964.14 0.320365
\(990\) −21722.8 −0.697368
\(991\) −16062.0 −0.514860 −0.257430 0.966297i \(-0.582876\pi\)
−0.257430 + 0.966297i \(0.582876\pi\)
\(992\) 92163.3 2.94979
\(993\) −6030.93 −0.192735
\(994\) 25013.2 0.798159
\(995\) −7861.07 −0.250465
\(996\) 43832.9 1.39448
\(997\) 1361.54 0.0432503 0.0216251 0.999766i \(-0.493116\pi\)
0.0216251 + 0.999766i \(0.493116\pi\)
\(998\) −96217.5 −3.05182
\(999\) −2657.44 −0.0841617
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 507.4.a.m.1.4 4
3.2 odd 2 1521.4.a.v.1.1 4
13.3 even 3 39.4.e.c.22.1 yes 8
13.5 odd 4 507.4.b.h.337.1 8
13.8 odd 4 507.4.b.h.337.8 8
13.9 even 3 39.4.e.c.16.1 8
13.12 even 2 507.4.a.i.1.1 4
39.29 odd 6 117.4.g.e.100.4 8
39.35 odd 6 117.4.g.e.55.4 8
39.38 odd 2 1521.4.a.bb.1.4 4
52.3 odd 6 624.4.q.i.529.1 8
52.35 odd 6 624.4.q.i.289.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.e.c.16.1 8 13.9 even 3
39.4.e.c.22.1 yes 8 13.3 even 3
117.4.g.e.55.4 8 39.35 odd 6
117.4.g.e.100.4 8 39.29 odd 6
507.4.a.i.1.1 4 13.12 even 2
507.4.a.m.1.4 4 1.1 even 1 trivial
507.4.b.h.337.1 8 13.5 odd 4
507.4.b.h.337.8 8 13.8 odd 4
624.4.q.i.289.1 8 52.35 odd 6
624.4.q.i.529.1 8 52.3 odd 6
1521.4.a.v.1.1 4 3.2 odd 2
1521.4.a.bb.1.4 4 39.38 odd 2